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Differentiable group actions on homotopy spheres. II. Ultrasemifree actions


Author: Reinhard Schultz
Journal: Trans. Amer. Math. Soc. 268 (1981), 255-297
MSC: Primary 57S15; Secondary 57R60, 57S25
DOI: https://doi.org/10.1090/S0002-9947-1981-0632531-6
MathSciNet review: 632531
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Abstract: A conceptually simple but very useful class of topological or differentiable transformation groups is given by semifree actions, for which the group acts freely off the fixed point set. In this paper, the slightly more general notion of an ultrasemifree action is introduced, and it is shown that the existing machinery for studying semifree actions on spheres may be adapted to study ultrasemifree actions equally well. Some examples and applications are given to illustrate how ultrasemifree actions (i) may be used to study questions not answerable using semifree actions alone, and (ii) provide examples of unusual smooth group actions on spheres with no semifree counterparts.


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  • [1] K. Abe, On semifree finite group actions on homotopy spheres, Publ. Res. Inst. Math. Sci. Kyoto Univ. 11 (1975), 267-280. MR 0400269 (53:4104)
  • [2] J. F. Adams and G. Walker, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 61 (1965), 81-103. MR 0171285 (30:1516)
  • [3] J. P. Alexander, G. C. Hamrick and J. W. Vick, Involutions on homotopy spheres, Invent. Math. 24 (1974), 35-50. MR 0345118 (49:9857)
  • [4] M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291. MR 0131880 (24:A1727)
  • [5] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II, Ann. of Math. (2) 88 (1968), 451-491. MR 0232406 (38:731)
  • [6] A. Bak, Surgery and $ K$-theory groups of quadratic forms over finite groups and orders, Universität Bielefeld, 1979 (preprint).
  • [7] J. C. Becker and R. E. Schultz, Equivariant function spaces and stable homotopy theory. I, Comment Math. Helv. 49 (1974), 1-34. MR 0339232 (49:3994)
  • [8] J. M. Boardman, Stable homotopy theory. Chapter VI: Duality and Thom spectra, Univ. of Warwick, 1966 (preprint).
  • [9] A. Borel (Editor), Seminar on transformation groups, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N. J., 1960. MR 0116341 (22:7129)
  • [10] G. Bredon, Introduction to compact transformation groups, Pure and Appl. Math., Vol. 46, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [11] -, Biaxial actions of classical groups, Rutgers Univ., New Brunswick, N. J., 1973 (preprint).
  • [12] W. Browder and T. Petrie, Diffeomorphisms of manifolds and semifree actions of homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 160-163. MR 0273636 (42:8513)
  • [13] E. H. Brown, Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965), 79-85. MR 0182970 (32:452)
  • [14] G. Brumfiel, [Free] differentiable $ {S^1}$ actions on homotopy spheres, Univ. of California, Berkeley, Calif., 1968 (preprint).
  • [15] S. Cappell and J. Shaneson, Linear algebra and topology, Bull. Amer. Math. Soc. (N. S.) 1 (1979), 685-687. MR 532553 (80f:57016)
  • [16] -, Nonlinear similarity of matrices, Bull. Amer. Math. Soc. (N.S) 1 (1979), 899-902. MR 546313 (80m:57010)
  • [17] -, Nonlinear similarity, Rutgers Univ. and Courant Inst, of Math. Sci., 1979 (preprint).
  • [18] -, Fixed points of periodic smooth maps, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), 5052-5054. MR 587279 (81m:57029)
  • [19] E. H. Connell, D. Montgomery and C.-T. Yang, Compact groups in $ {E^n}$, Ann. of Math. (2) 80 (1967), 94-103; correction, ibid. 81 (1965), 194.
  • [20] K. O. Dahlberg, $ d$-pseudofree actions on homotopy spheres, Ph.D. Thesis, Princeton Univ., Princeton, N. J., 1976.
  • [21] T. tom Dieck, The Burnside ring and equivariant stable homotopy theory (notes by M. Bix), Univ. of Chicago, 1975 (mimeographed). MR 0423389 (54:11368)
  • [22] -, Transformation groups and representation theory, Lecture Notes in Math., vol. 766, Springer, New York, 1980.
  • [23] A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. (2) 98 (1963), 223-255. MR 0155330 (27:5264)
  • [24] -, Halbexakte Homotopiefunktoren, Lecture Notes in Math., vol. 13, Springer, New York, 1966. MR 0198464 (33:6622)
  • [25] J. E. Ewing, Spheres as fixed point sets, Quart. J. Math. Oxford Ser. (2) 27 (1976), 445-455. MR 0431233 (55:4234)
  • [26] -, Semifree actions of finite groups on homotopy spheres, Trans. Amer. Math. Soc. 245 (1978), 431-442. MR 511421 (80a:57020)
  • [27] S. Feder and S. Gitler, Stunted projective spaces and the $ J$-order of the Hopf bundle, Bull. Amer. Math. Soc. 80 (1974), 748-749. MR 0348736 (50:1233)
  • [28] H. Hauschild, Allgemeine Lage und äquivariante Homotopie, Math. Z. 143 (1975), 155-164. MR 0385899 (52:6758)
  • [29] -, Äquivariante Whiteheadtorsion, Manuscripta Math. 26 (1978), 63-82. MR 513146 (80g:57023)
  • [30] W.-C. Hsiang and W.-Y Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 750-786. MR 0217213 (36:304)
  • [31] W. Iberkleid, Pseudo linear spheres, Michigan Math. J. 25 (1978), 359-370.
  • [32] S. Illman, Equivariant algebraic topology, Ph.D. Thesis, Princeton Univ., Princeton, N. J., 1972. MR 0358761 (50:11220)
  • [33] -, Recognition of linear actions on spheres, Univ. of Helsinki, 1979 (preprint).
  • [34] K. Jänich, On the classification of $ G$-manifolds, Math. Ann. 176 (1968), 53-76.
  • [35] F. E. A. Johnson, Triangulation of stratified sets and other questions in differential topology, D. Phil. Thesis, Univ. of Liverpool, 1972.
  • [36] L. E. Jones, The converse to the fixed point theorem of P. A. Smith. II, Indiana Univ. Math. J. 22 (1972), 309-325. MR 0339249 (49:4010)
  • [37] K. Kawakubo, Weyl group actions and equivariant homotopy equivalence, Proc. Amer. Math. Soc. 80 (1980), 172-176. MR 574530 (81j:57022)
  • [38] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres, Ann. of Math. (2) 78 (1963), 514-527. MR 0148075 (26:5584)
  • [39] H.-T. Ku and M.-C. Ku, Semifree differentiable actions of $ {S^1}$ on homotopy $ (4k + 3)$-spheres, Michigan Math. J. 15 (1968), 471-476. MR 0239622 (39:979)
  • [40] R. Lee, Differentiable classification of some topologically linear actions, Bull. Amer. Math. Soc. 75 (1969), 441-444. MR 0239623 (39:980)
  • [41] W. Lellmann, Orbiträume von $ G$-Mannigfaltigkeiten und stratifizierte Mengen, Diplomarbeit, Universität Bonn, 1975.
  • [42] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15-50. MR 0180981 (31:5211)
  • [43] -, Unknotting spheres in codimension two, Topology 4 (1965), 9-16. MR 0179803 (31:4045)
  • [44] -, Semifree circle actions on spheres, Invent. Math. 22 (1973), 161-186. MR 0350767 (50:3259)
  • [45] P. Löffler, Homotopielineare $ {{\mathbf{Z}}_p}$-Operationen auf Sphären, Topology 20 (1981), 291-312. MR 608602 (83b:57024)
  • [46] M. Mahowald, The metastable homotopy of $ {S^n}$, Mem. Amer. Math. Soc. No. 72 (1967). MR 0236923 (38:5216)
  • [47] -, A new infinite family in $ _2\pi _{\ast}^S$, Topology 16 (1977), 249-256. MR 0445498 (56:3838)
  • [48] M. Mahowald, Some homotopy classes generated by the $ {\eta _j}$, Algebraic Topology--Aarhus 1978 (Proc. 50th Anniv. Conf.), Lecture Notes in Math., vol. 763, Springer, New York, 1979, pp. 23-37. MR 561212 (83e:55008)
  • [49] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349-369. MR 0214072 (35:4924)
  • [50] T. Matumoto, Equivariant $ K$-theory and Fredholm operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363-374. MR 0290354 (44:7538)
  • [51] -, On $ G$-$ CW$ complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363-374. MR 0345103 (49:9842)
  • [52] -, Equivariant cohomology theories on $ G$-$ CW$ complexes, Osaka J. Math. 10 (1973), 51-68. MR 0343259 (49:8003)
  • [53] J. P. May, J. McClure and G. Triantafillou, Equivariant localizations, Univ. of Chicago, 1981 (preprint).
  • [54] H. Miller, D. Ravenel and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), 469-516. MR 0458423 (56:16626)
  • [55] J. W. Milnor and J. D. Stasheff, Characteristic classes, Ann. of Math. Studies, No. 76, Princeton Univ. Press, Princeton, N. J., 1974. MR 0440554 (55:13428)
  • [56] D. Montgomery and C.-T. Yang, Differentiable group actions on homotopy seven spheres. I, Trans. Amer. Math. Soc. 133 (1967), 480-498. MR 0200934 (34:820)
  • [57] S. Oka, The stable homotopy groups of spheres. I. Hiroshima Math. J. 1 (1971), 305-337. MR 0310879 (46:9977)
  • [58] P. Orlik, Homotopy $ 4$-spheres have little symmetry, Math. Scand. 33 (1973), 275-278. MR 0346829 (49:11551)
  • [59] C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1-26. MR 0090053 (19:761a)
  • [60] W. Pulikowski, $ RO(G)$-graded $ G$-bordism theory, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom Phys. 21 (1973), 991-995. MR 0339234 (49:3996a)
  • [61] D. Ravenel, The cohomology of the Morava stabilizer algebras, Math. Z. 152 (1977), 287-297. MR 0431168 (55:4170)
  • [62] -, A novice's guide to the Adams-Novikov spectral sequence, Geometric Applications of Homotopy Theory. II (Proc. Conf., Evanston, 1977), Lecture Notes in Math., vol. 658, Springer, New York, 1978, pp. 404-475.
  • [63] M. Rothenberg, Differentiable group actions on spheres, Proc. Advanced Study Inst. Algebraic Topology (Aarhus, 1970), Math. Inst. Aarhus Univ., 1970, pp. 455-475. MR 0301764 (46:919)
  • [64] -, Torsion invariants and finite transformation groups, Proc. Sympos. Pure Math., vol. 32, Part 1, Amer. Math. Soc., Providence, R. I., 1978, pp. 267-311. MR 520507 (80j:57038)
  • [65] M. Rothenberg and J. Sondow, Nonlinear smooth representations of compact Lie groups, Pacific J. Math. 84 (1979), 427-444. MR 568660 (81d:57033)
  • [66] R. Schultz, Circle actions on homotopy spheres bounding plumbing manifolds, Proc. Amer. Math. Soc. 36 (1972), 297-300. MR 0309138 (46:8248)
  • [67] -, Homotopy decompositions of equivariant function spaces. I: Spaces of principal bundle maps, Math. Z. 131 (1973), 49-75. MR 0407866 (53:11636)
  • [68] -, Homotopy decompositions of equivariant function spaces. II: Function spaces of equivariantty triangulated spaces, Math. Z. 132 (1973), 69-90. MR 0407867 (53:11637)
  • [69] -, Homotopy sphere pairs admitting semifree differentiable actions, Amer. J. Math. 96 (1974), 308-323. MR 0368053 (51:4295)
  • [70] -, Rational $ h$-cobordism invariants for lens space bundles, Quart. J. Math. Oxford. Ser. (2) 25 (1974), 497-512; correction, ibid. 28 (1977), 128.
  • [71] -, Differentiable $ {{\mathbf{Z}}_p}$ actions on homotopy spheres, Bull. Amer. Math. Soc. 80 (1974), 961-964. MR 0356105 (50:8576)
  • [72] -, Differentiable group actions on homotopy spheres. I: Differential structure and the knot invariant, Invent. Math. 31 (1975), 105-128. MR 0405471 (53:9264)
  • [73] -, Equivariant function spaces and equivariant stable homotopy theory, Transformation Groups (Proc. Newcastle-upon-Tyne Conf., 1976), London Math. Soc. Lecture Notes No. 26, Cambridge Univ. Press, New York, 1977, pp. 169-189. MR 0494172 (58:13097)
  • [74] -, On the topological classification of linear representations, Topology 16 (1977), 263-269. MR 0500964 (58:18449)
  • [75] -, Spherelike $ G$-manifolds with exotic equivariant tangent bundles, Studies in Algebraic Topology (Adv. in Math. Supp. Studies, Vol. 5), Academic Press, New York, 1979, pp. 1-39.
  • [76] -, Smooth actions of small groups on exotic spheres, Proc. Sympos. Pure Math., Vol. 32, Part 1, Amer. Math. Soc., Providence, R. I., 1978, pp. 155-160. MR 520503 (80e:57053)
  • [77] -, Isotopy classes of periodic diffeomorphisms on spheres, Algebraic Topology, (Proc. Conf. Waterloo, 1978), Lecture Notes in Math., vol. 741, Springer, New York, 1979, pp. 334-354. MR 557176 (81i:57030)
  • [78] M. Sebastiani, Sur les actions à deux points fixes des groupes finis sur les sphères, Comment. Math. Helv. 45 (1970), 405-439. MR 0278337 (43:4067)
  • [79] G. Segal, Equivariant $ K$-theory, Inst. Hautes Études Sci. Publ. Math. No. 34 (1968), 129-151. MR 0234452 (38:2769)
  • [80] -, Equivariant stable homotopy theory, Actes, Congrès Internat. Math. (Nice, 1970), T.2, Gauthier-Villars, Paris, 1971, pp. 59-63. MR 0423340 (54:11319)
  • [81] J. Shaneson, Embeddings with codimension 2 of spheres in spheres and $ h$-cobordisms of $ {S^1} \times {S^3}$, Bull. Amer. Math. Soc. 74 (1968), 972-974. MR 0230325 (37:5887)
  • [82] -, Wall's surgery obstruction groups for $ G \times {\mathbf{Z}}$, Ann. of Math. (2) 90 (1969), 296-334. MR 0246310 (39:7614)
  • [83] L. C. Siebenmann, Topological manifolds, Actes, Congrès Internat. Math. (Nice, 1970), T.2, Gauthier-Villars, Paris, 1971, pp. 133-163. MR 0423356 (54:11335)
  • [84] L. Smith, Realizing complex bordism modules. IV, Amer. J. Math. 99 (1977), 418-436. MR 0433450 (55:6426)
  • [85] J. Stallings, On topologically unknotted spheres, Ann. of Math. (2) 77 (1963), 490-503. (Fourth Ed., Publish or Perish, Berkeley, Calif., 1977.) MR 0149458 (26:6946)
  • [86] D. Sullivan, Genetics of homotopy theory and Adams conjecture, Ann. of Math. (2) 100 (1974), 1-79. MR 0442930 (56:1305)
  • [87] M. Tangora, Some homotopy groups $ \bmod 3$, Reunion Sobre Teoria de Homotopia (Northwestern, 1974), Notas Mat. Simposia No. 1, Soc. Mat. Mexicana, México, D. F., 1975, pp. 227-246. MR 761731
  • [88] C. T. C. Wall. Classification of Hermitian forms. VI: Group rings, Ann. of Math. (2) 103 (1976), 1-80. MR 0432737 (55:5720)
  • [89] K. Wang, Semifree actions on homotopy spheres, Trans. Amer. Math. Soc. 211 (1975), 321-337. MR 0377951 (51:14120)
  • [90] J. A. Wolf, Spaces of constant curvature, 4th Ed., Publish or Perish, Berkeley, Calif., 1977.
  • [91] W. S. Massey, On the normal bundle of a sphere in Euclidean space, Proc. Amer. Math. Soc. 10 (1959), 959-964. MR 0109351 (22:237)
  • [92] W. Browder, Surgery and the theory of differentiable transformation groups, Proc. Conf. on Transformation Groups (New Orleans, 1967), Springer, New York, 1968, pp. 1-46. MR 0261629 (41:6242)
  • [93] H. B. Lawson and S.-T. Yau, Scalar curvature, nonabelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helv. 49 (1974), 232-244. MR 0358841 (50:11300)
  • [94] J. F. Adams, Operation of the nth kind in homotopy theory, and what we don't know about $ {\mathbf{R}}{P^\infty }$, New Developments in Topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), London Math. Soc. Lecture Notes No. 11, Cambridge Univ. Press, New York, 1974, pp. 1-9. MR 0339178 (49:3941)
  • [95] S. Illman, Smooth triangulation of $ G$-manifolds for $ G$ a finite group, Math. Ann. 233 (1978), 199-220. MR 0500993 (58:18474)
  • [96] G. Bredon, Equivariant stable stems, Bull. Amer. Math. Soc. 73 (1967), 269-273. MR 0206947 (34:6763)
  • [97] -, Equivariant homotopy, Proc. Conf. on Transformation Groups (New Orleans, 1967), Springer, New York, 1968, pp. 281-292. MR 0250303 (40:3542)
  • [98] W. Browder and W.-C. Hsiang, Some problems in homotopy theory, manifolds, and transformation groups, Proc. Sympos. Pure Math., vol. 32, Part 2, Amer. Math. Soc., Providence, R. I., 1978, pp. 251-267. MR 520546 (80e:55001)
  • [99] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math., vol. 347, Springer, New York, 1973. MR 0420609 (54:8623a)
  • [100] H. Hauschild, J. P. May and S. Waner, Equivariant infinite loop space theory (to appear).
  • [101] J. P. May, $ {E_\infty }$ ring spaces and $ {E_\infty }$ ring spectra, Lecture Notes in Math., vol. 577, Springer, New York, 1977. MR 0494077 (58:13008)
  • [102] A. Verona, Triangulation of stratified fibre bundles, Manuscripta Math. 30 (1980), 425-445. MR 567218 (81f:57019)

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DOI: https://doi.org/10.1090/S0002-9947-1981-0632531-6
Article copyright: © Copyright 1981 American Mathematical Society

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